Boundary Conditions
Monte Carlo and Molecular Dynamics simulations aim to provide
information about the properties of a macroscopic sample (by simulating atoms and molecules and then using the theory of statistical mechanics to bridge from microscopic to macroscopic information)
The number of atoms that can be conveniently handled in present-day computers ranges from a few hundred to a few million. Most simulations probe the structural and thermodynamic properties of a system of a few thousand particles. For such small systems the choice of boundary conditions can strongly affect the measured properties.
For free boundaries, the fraction of all molecules that are at the surface is proportional to N-1/3. For example, in a
Boundary Conditions
For most studies, you don’t want to study a nano-sized droplet of water! Instead, you would like to study a “piece” of a large region of liquid water.
Why? Because the properties of water (and other molecules) at an interface are very different from in the bulk For example, water's surface is acidic. The hydronium ion prefers to sit at the water surface rather than deep inside the liquid. Whereas H2O
molecules typically form four hydrogen bonds to their neighbours, H3O+ can only form three. The three hydrogens can bind to other water molecules, but the oxygen atom, where most of the positive charge resides, can no longer act as a good 'acceptor' for hydrogen bonds.
So, hydronium acts somewhat like an amphiphile - a molecule with a
water-soluble part and a hydrophobic part, like a soap molecule. The ions gather at the air-water interface with the hydrogen atoms pointing
downwards to make hydrogen bonds and the oxygen pointing up out of the liquid. In 2005 Saykally and his colleague Poul Petersen used a
Boundary Conditions
For most studies, you don’t want to study a nano-sized droplet of water! Instead, you would like to study a “piece” of a large region of liquid water.
Boundary Conditions
Snapshots (side view) of thesolution–air interface of 1.2 M aqueous sodium halides and
density profiles (number densities) of water oxygen atoms and ions plotted vs. distance from the center of the slabs in the direction normal to the interface, normalized by the bulk water density. From top to
bottom the systems are NaF, NaBr, and NaI. The colors of the density profiles correspond to the coloring of the atoms in the snapshots (blue for water and green for Na+ in all of the plots, black for F, orange for Br, and magenta for I). The
water density is scaled differently from those of the ions so that it can be easily displayed on the same
Boundary Conditions
Say you want to study ion solvation
WITHOUT interface effects. But you can only afford to simulation a few thousand atoms. What should you do?
Finite size effects: phase transitions
Ehrenfest classification scheme: group phase transitions based on the degree of non-analyticity involved.
Under this scheme, phase transitions are labeled by the lowest derivative of the free energy that is discontinuous at the transition.
First-order phase transitions exhibit a discontinuity in the first derivative of the free energy with a thermodynamic variable. The various solid/liquid/gas transitions are classified as first-order
transitions because they involve a discontinuous change in density (which is the first derivative of the free energy with respect to chemical potential.)
Finite size effects: phase transitions
But there is a problem: with a finite number of variables all state functions are analytic (no discontinuities).
You have to do simulations with increasingly larger systems and look for where the quantity of interest begins to diverge.
J. Phys. A: Math. Gen. 35 (2002) 33–42
C
V=
∂U
∂T
⎛
⎝
⎜
⎞
Molecular Dynamics Simulations
Molecular Dynamics simulations are in many respects similar to real experiments.
When we perform a real experiment, we proceed as follows: 1)Prepare sample of the material for study 2)Connect the sample to a measuring device (e.g. thermometer) 3)Measure property of interest over a certain time interval (the longer we measure or the more times we repeat the measurement, the more accurate our measurement becomes due to statistical noise) For MD we follow the same approach. 1) first we prepare a sample by selecting a system of N atoms 2) we then solve Newton’s equations of motion for this system until the system properties no longer change with time (we equilibrate the system); 3) after equilibration we perform the actual measurement.
Molecular Dynamics Simulations
If we solve Newton’s equations, we (in principle) conserve energy. But this is not compatible with sampling configurations from the Boltzmann
distribution. So how do we make sure we are visiting configurations in accordance with their Boltzmann weight?
One way to do this is to use the Anderson thermostat. In this method, the system is coupled to a heat bath that imposes the desired temperature.
T
=
2
3
k
B1
N
1
2
m
iv
i2 i=1N
∑
The coupling to a heat bath is represented by stochastic impulsive forces that act occasionally on randomly selected particles. These stochastic
collisions with the heat bath can be considered as Monte Carlo moves that transport the system from one constant-energy shell to another.
Molecular Dynamics Simulations
We take the distribution between time intervals for two successive stochastic collisions to be P(t;v) which is of Poisson form
A constant temperature simulation consists of 1)Start with an initial set of positions and momenta and integrate the
equations of motion for a time Δt. 2)A number of particles are selected to undergo a collision with the heat bath. The probability that a particle is selected in a time step of length Δt is vΔt. 3)If particle i has been selected to undergo a collision, its new velocity is
drawn from a Maxwell-Boltzmann distribution corresponding to the desired temperature T. All other particles are unaffected by this collision.
P
(
t
;
v
) =
v
exp −
(
vt
)
Where P(t;v)dt is the probability that the next collision will take place inStatistical mechanics of non-equilibrium
systems: linear response theory and the
fluctuation-dissipation theorem
Up to this point, we have used statistical mechanics to examine equilibrium properties (the Boltzmann factor was derived assuming equilibrium).
What about non-equilibrium properties?
1968 Nobel Prize in chemistry: Lars Onsager
Example: relaxation rate by which a system reaches equilibrium from a prepared non-equilibrium state (e.g. begin with all reactions, no products, and allow a chemical reaction to proceed)
Our discussion will be limited to systems close to equilibrium. In this
regime, the non-equilibrium behavior of macroscopic systems is described by linear response theory.
Systems close to equilibrium
What do we mean by “close” to equilibrium?
We mean that the deviations from equilibrium are linearly related to the perturbations that remove the system from equilibrium. For example, consider an aqueous electrolyte solution. At equilibrium, there is no net flow of charge; the average current <j> is zero.
At some time t = t1, an electric field of strength E is applied, and the charged ions begin to flow. At time t = t2, the field is then turned off. Let j(t) denote the observed current as a function of time. The non-equilibrium behavior, j(t) ≠ 0, is linear if j(t) is proportional to E.
Systems close to equilibrium
A final example: to derive the selection rules for Raman spectroscopy we assume that the applied electric field (namely shining light on a molecule) inducing a dipole in the molecule in a manner such that the induced dipole is proportional (linearly related) to the applied field strength. The
Onsager’s regression hypothesis
and time correlation functions
When left undisturbed, a non-equilibrium system will relax to its
thermodynamic equilibrium state. When not far from equilibrium, the relaxation will be governed by a principle first proposed in 1930 by Lars Onsager in his regression hypothesis:
The relaxation of macroscopic non-equilibrium disturbances is governed by the same laws as the regression of spontaneous microscopic fluctuations in an equilibrium system.
This hypothesis is an important consequence of a profound theorem in mechanics: the fluctuation-dissipation theorem.
To explore this hypothesis, we need to talk about correlations of spontaneous fluctuations, which is done using time correlation functions.
A
(
t
) =
A
(
t
) −
A
Define as the instantaneous deviation (fluctuation) of A from its equilibrium average
Onsager’s regression hypothesis
and time correlation functions
Of course,
However, if one looks at equilibrium correlations between fluctuations at different times, one can learn something about the system
C
(
t
) =
δ
A
(0)
δ
A
(
t
) =
A
(0)
A
(
t
) −
A
2
A
= 0
The correlation between and an instantaneous or spontaneous fluctuation at time zero is
A
(
t
)
Look at the limiting behavior: at short times,
C
(0) =
( )
δ
A
2while at long times is uncorrelated to , thus
A
(
t
)
A
(0)
C
(
t
)
→
δ
A
(0)
δ
A
(
t
) = 0 as
t
→
∞
Onsager’s regression hypothesis
and time correlation functions
By invoking the ergodic hypothesis, we can view the equilibrium average as a time average, and write
“sliding window”
A
(
t
)
Δ
A
(0)
=
C
(
t
)
C
(0)
A
(0)
δ
A
(
t
) = lim
T→∞
1
T
0d
τ δ
A
(
τ
)
δ
A
(
τ
+
t
)
T
∫
We can now state Onsager’s regression hypothesis. Imagine at time t=0, we prepare a system in a non-equilibrium state and allow it to relax to
equilibrium. In the linear regime, the relaxation obeys
where
A
(
t
) =
A
(
t
) −
A
C
(
t
) =
δ
A
(0)
δ
A
(
t
)
In a system close to equilibrium, we cannot distinguish between spontaneous fluctuations and deviations from equilibrium that are externally prepared. Since we cannot distinguish, the relaxation of should indeed
coincide with the decay to equilibrium of
A
(
t
)
Application: (self-) diffusion
Fick’s 1st Law: the diffusive flux goes from regions of high concentration to
regions of low concentration, with a magnitude that is proportional to the concentration gradient
J is the flux: the amount of substance that will flow through a small area in a small time interval
J
= −
D
∇
φ
D is the diffusion constant
ϕ is the concentration
Fick’s 2nd Law: use the 1st law and mass conservation to get
t
=
D
∇
2
φ
Application: (self-) diffusion
The constant D is called a transport coefficient. To learn how this transport coefficient is related to the microscopic dynamics, let us consider the
correlation function
Where is the instantaneous density at position r and time t
C
(
r
,
t
) =
δρ
(
r
,
t
)
δρ
(0,0)
(
r
,
t
)
(r,t) = δ[r−r j(t)] j=1
N
∑
According to Onsager’s regression hypothesis, C(r,t) also obeys Fick’s 2nd law
C
(
r
,
t
)
∂t
=
D
∇
2
C
(
r
,
t
)
(
r
,
t
)
ρ
(0,0)
Notice that is proportional to
Application: (self-) diffusion
Why is this? First, recall that
C
(
r
,
t
) =
δρ
(0,0)
δρ
(
r
,
t
) =
ρ
(0,0)
ρ
(
r
,
t
) −
ρ
2As a result, we have
P
(
r
,
t
)
∂t
=
D
∇
2
P
(
r
,
t
)
Now, consider which is the mean squared displacement of a tagged solute molecule in a time t.
R
2(
t
) =
r
1
(
t
) −
r
1(0)
2Clearly,
R
2(
t
) =
∫
dr r
2P
(
r
,
t
)
Hence,
d
dt
Δ
R
2
(
t
) =
dr r
2∂P
(
r
,
t
)
∂t
Application: (self-) diffusion
is normalized for any value of time, giving
We can write
P
(
r
,
t
)
d
dt
Δ
R
2
(
t
) = 6
D
R
2(
t
) = 6
Dt
or,
This formula was first derived by Einstein
Notice that ballistic motion looks like
R
(
t
)
∝
t
r
1(
t
) −
r
1(0) =
d
τ
v
(
τ
)
0t
∫
where v(t) is the particle velocity
R
2(
t
) =
r
1
(
t
) −
r
1(0)
2=
d
τ
0t
∫
d
φ
v
(
τ
)
⋅
v
(
φ
)
0t
∫
HenceTake d/dt on both sides
d
dt
Δ
R
2
(
t
) = 2
d
τ
0
t
Application: (self-) diffusion
giving
Since the left-hand side goes to 6D in the limit of large times, we have
2
d
τ
0t
∫
v
(
t
)
⋅
v
(
τ
) = 2
v
(
t
)
⋅
[
r
(
t
) −
r
(0)
] = 2
v
(0)
⋅
[
r
(0) −
r
(−
t
)
]
2
d
τ
−t0
∫
v
(0)
⋅
v
(
τ
) = −2
d
τ
0 −t∫
v
(0)
⋅
v
(
τ
) = 2
d
τ
0t
∫
v
(0)
⋅
v
(−
τ
)
d
dt
Δ
R
2
(
t
) = 2
d
τ
0
t
∫
v
(0)
⋅
v
(
τ
)
D
=
13
d
τ
0 ∞∫
v
(0)
⋅
v
(
τ
)
